Research Article | Open Access
The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations
We study the stability issue of the generalized 3D Navier-Stokes equations. It is shown that if the weak solution of the Navier-Stokes equations lies in the regular class , , , , then every weak solution of the perturbed system converges asymptotically to as , .
1. Introduction and Main Result
In this study, we consider the Cauchy problem of the generalized 3D Navier-stokes equations: Here, , and and denote unknown velocity and pressure, respectively. is the external force and is a given initial velocity.
It is well known that when , system (1) becomes the classic Navier-Stokes equations. For the Navier-Stokes equations, it is proved that it has a global weak solution for given with . However, the regularity of Leray weak solutions is still an open problem in mathematical fluid mechanics even if much effort has been made [2–4]. It is an interesting problem to investigate the stability properties of the Navier-Stokes equations and related fluid models [5–11]. As regard to the above system (1), the asymptotic stability of weak solution of the generalized 3D Navier-Stokes equation is described as follows. If is perturbed initially by without any smallness assumption, then the perturbed system is governed by the following equations: where is the initial perturbation. There is large literature on the stability issue of the classic Navier-Stokes equations and related fluid models [12–17]. The aim of this paper is to show the stability of weak solution in the framework of the homogeneous Besov space. More precisely, with the use of the Littlewood-Paley decomposition and the classic Fourier splitting technique, we can show that when the initial perturbation , then every weak solution of the perturbed system (2) converges asymptotically to as .
Now our result reads as follows.
The remainder of this paper is organized as follows. In the Section 2, we first recall the Littlewood-Paley decomposition and the Bony decomposition; then we give three key lemmas. And we prove asymptotic stability of the weak solution in the Section 3.
2. Some Auxiliary Lemmas
We recall some basic facts about the Littlewood-Paley decomposition (refer to ). Let be Schwartz class of rapidly decreasing functions; supposing , the Fourier transformation is defined by Choose two nonnegative radial functions , supported in and , respectively, such that
Let and , we define the dyadic blocks as follows: We can easily verify that Especially for any , we have the Littlewood-Paley decomposition:
Now we give the definition of the Besov space. Let and ; the inhomogeneous Besov space (see ) is defined by the full-dyadic decomposition, such as where and is a dual space of .
The Bony decomposition (see ) will be frequently used; it is followed by where
The following Bernstein inequality (see ) will be used in the next section.
Lemma 2. Assume that and , for , one has and the constant is independent of and .
In the following, we will introduce two lemmas, which will be employed in the proof of our theorem.
Lemma 3. Suppose that , for all , , , .
Then the trilinear form is continuous and In particular, if , then
Proof of Lemma 3. We borrow the idea of  to prove this lemma. By using of the Littlewood-Paley decomposition and the Bony decomposition, we obtain
Then we estimate , , and one by one. Applying the Hölder inequality and the Bernstein inequality (40), we derive where .
Since and with , then
Thanks to the Sobolev embedding , we have the following estimate:
Similarly, for , we also have
To estimate the last term , by using the Hölder inequality and the Bernstein inequality we obtain Since and , we have
So, we can derive
Applying the interpolation inequality, we have
Especially if , by using the interpolation inequality, we get Hence, the proof of the lemma is complete.
Lemma 4. Let be the solution of the above problem. Then
Proof of Lemma 4. Taking the Fourier transformation of the first equation of (38), we get
We can easily obtain Applying the operator to the first equation of (38), we have and taking the Fourier transformation, we get thus
Then we have Thus solving the ordinary differential equation (31) and using (36) gives which is the desired assertion of Lemma 4.
3. Proof of Theorem 1
Taking the inner product of the first equation in (38) with together with the divergence-free condition of we have
Applying Plancherel’s theorem to (38) yields
Let be a continuous function of with , and , we can derive the following:
By integrating in time from to for (40), we have
Noting that is a scalar function and applying Lemma 3, we get
Let , we have
Choose , then
By using the Gronwall inequality, it follows that
Since we derive which completes the proof of Theorem 1.
The authors want to express their sincere thanks to the editor and the referees for their invaluable comments and suggestions. This work is partially supported by the NNSF of China (11271019), NSF of Anhui Province (11040606M02) and is also financed by the 211 Project of Anhui University (KJTD002B, KJJQ005).
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Copyright © 2013 Wen-Juan Wang and Yan Jia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.